Reference no: EM133116137

Assessment Requirements

You should complete this work in CVX and write a report (either in LaTeX or Word) containing main findings and brief comments/justification for every task. Your submission should be one document that includes both your report and CVX scripts. This coursework is an‘element of assessment' which means you need to pass it in order to pass the module.

Assessment Scenario/Problem

You have to carry out the following two tasks.

Task 1
Assume that you work in a computerised tomography (CT) laboratory and your task is to reconstruct images of human brain phantoms. However, the reconstructed images that you get from the scanner are corrupted with additive noise, and you are asked by your boss to come up with a strategy to de- corrupt them. You decide that you will solve the problem using a smoothing regulariser.

(a) What is the price/unwelcome experience that will be undergone for achieving denoising?

(b) Suppose that the image that you want to reconstruct is the modified Shepp-Logan phantom of the human brain with a spatial image resolution of 200x200. You can obtain this image in the CVX environment by typing in
P = phantom('Modified Shepp-Logan',200);

Alternatively, you can obtain the same image by reading (using the ‘load' command) the related ‘.mat' file that I have uploaded to NOW. Plot this image. Of course, you don't know the exact image that you are trying to reconstruct but you have an approximate knowledge of how human brain phantoms looks like. Which regulariser would you choose for your denoising task? Justify your answer.

(c) Assume that the corrupted version of the image that you get from the scanner, Pn, is equivalent to the true image, P, after adding Gaussian noise with zero mean and standard deviation equal to 0.06. Create and illustrate this noisy version. This is the image that you will work with.

(d) If the reconstructed image pixel intensities Prec(i,j) (for i, j = 1, ..., 200) need to be between 0 and 1 (0 ≤ Prec(i,j) ≤ 1), formulate the image reconstruction problem as a scalarised bi-criterion convex optimisation problem. Then, load it to CVX/CVXPY and solve it for three distinct levels of smoothing. Your chosen three levels should approximately correspond to ‘too much', ‘about right', and ‘not quite enough' smoothing. Provide the solutions to the three optimisation problems, that is, the illustrations of the reconstructed images.

(e) Plot the trade-off curve for this bi-criterion optimisation problem. In the plot show the Pareto optimal point that you will eventually use.

(f) Provide all the CVX/CVXPY code used.

Task 2
One-Vs-All classification is a strategy that leverages binary algorithms for multi-class classification. It involves fitting one binary classifier per output class. For each classifier, the class is fitted against all the other classes. Finally, given a new input point, predictions are made by running all binary classifiers, and choosing the output class that corresponds to the prediction with the highest confidence.

Suppose you have been asked to build a model that best maps sepal and petal lengths to three types of roses, namely hybrid tea, grandiflora, and floribunda. That is to say, the explanatory variables comprise sepal length and petal length, whereas the output classes are the three types of roses. The dataset for this task can be found in the file ‘roses.m', and it contains 20 instances of each rose type, where the input features have been normalised.

Assume that you decided to deal with this task by using convex optimisation. In particular, you have chosen to employ the One-Vs-All strategy and the logistic model

p = prob(y = 1) = exp(a^Tu + b) / [1 + exp(a^Tu + b)]

where a ∈ Rⁿ and b ∈ R are the model parameters, u ∈ Rⁿ contains the explanatory variables, and y equals 1 means that an individual rose from the population belongs to a certain type.

(a) Suppose the whole rose population in the dataset is to be used for fitting each binary classifier. Provide the formulation of the convex optimisation problem that you will use for each binary classifier. Explain all notation used and why this problem is convex.

(b) Use CVX/CVXPY software to calculate the maximum likelihood estimates of the model parameters a, b for each binary classifier, and present the results.

(c) Assume that you have been given a rose with normalised sepal and petal lengths (0,0)T. Use the model you built above to predict the class (rose type) to which it belongs.

(d) Provide all the CVX/CVXPY code used.